MHB Hyperreal Intermediate Value Theorem

[poissonspot]

I haven't thought too much about this, but it seems to me that the intermediate value theorem would transfer. Am I incorrect?

 

[Plato2]

I haven't thought too much about this, but it seems to me that the intermediate value theorem would transfer. Am I incorrect?

In Elementary Caculus: An Infinitesimal Approach by Jerome Keisler in chapter 3, there is good proof of this.
The chapters and whole book is a free down-load HERE.

 

[poissonspot]

In Elementary Caculus: An Infinitesimal Approach by Jerome Keisler in chapter 3, there is good proof of this.
The chapters and whole book is a free down-load HERE.

This was not quite what I was thinking about. But I think it helps clarify the question I had. I have not thought much about the topology of the hyperreals but what I'm asking is:

Say I have a continuous function defined on the hyperreals and I take a "closed" interval [a,b], that is the set of all hyperreals between a and b, where a and b may very well be unlimited, does it follow that for any c: f(a)<c<f(b) there exists an x so that f(x)=c.

The star transform of [a,b] wouldn't give unlimited elements as far I'm looking into it, so I'm afraid this would have to be proven using something other than Los's theorem.